Combine 2 cups milk and onions in a large bowl. Add bread, tossing gently to coat. Cover and chill for 30 minutes. Stir in spinach.
Preheat oven to 375°.
Heat the oil in a large nonstick skillet over medium-high heat. Add mushrooms; sauté 5 minutes. Add bell pepper; sauté 3 minutes. Add prosciutto and garlic; sauté 1 minute. Remove from heat; stir in 1 tablespoon basil, 1/8 teaspoon salt, and 1/8 teaspoon black pepper.
Place 1 cup milk, 2 tablespoons basil, 1/8 teaspoon salt, 1/8 teaspoon black pepper, eggs, and egg whites in a large bowl; stir well with a whisk.
Place half of bread mixture in an 11 x 7-inch dish coated with cooking spray. Spoon mushroom mixture over bread mixture, and sprinkle with half of cheese. Top with remaining bread mixture. Pour egg mixture over bread mixture; sprinkle with remaining cheese. Bake at 375° for 1 hour or until set.